# dz + η 1 r r 2 + c 1 ln r + c 2 subject to the boundary conditions of no-slip side walls and finite force over the fluid length u z at r = 0

Size: px
Start display at page:

Download "dz + η 1 r r 2 + c 1 ln r + c 2 subject to the boundary conditions of no-slip side walls and finite force over the fluid length u z at r = 0"

Transcription

1 Poiseuille Flow Jean Louis Maie Poiseuille, a Fench physicist and physiologist, was inteested in human blood flow and aound 1840 he expeimentally deived a law fo flow though cylindical pipes. It s extemely useful fo all kinds of hydodynamics such as plumbing, flow though hypedemic needles, flow though a dinking staw, flow in a volcanic conduit, etc. Fo this eason, it is geneally known as pipe flow. Actually, the cgs unit of viscosity, the Poise P, was named afte Poiseuille and is still used in many engineeing texts. A single Poise is equivalent to 10 Pa-s the SI unit, thus making the Pa-s measue of viscosity the peso of fluid dynamics and Poise equivalent to the dolla. This is the fist of many special cases of Navie-Stokes equation in which vey simplified situations can be solved analytically. Pipe flow is defined to be unidiectional, i.e. thee is only a single non-zeo component of velocity and that component is both independent of distance in the flow diection and has the same diection eveywhee. The geomety is that of a long cylindical pipe with length l and adius a so the appopiate coodinate system is cylindical pola, θ, z. The pessues at each end of the pipe ae P 1 and P 0 so the pessue gadient, dp/, is constant eveywhee in the pipe. The unidiectional natue of the poblem means u and u θ, thus the continuity equation is educed to uz. This means that because z of the incompessibility constaint, at any value of z the velocity must both be a constant value as well as have an identical velocity pofile. Futhemoe, any change in the flow will occu eveywhee in the pipe instantaneously. Of couse, you aleady know this is tue because you have taken a showe without a pessue egulato so that when somebody else flushes a toilet and cold wate is diveted to efill the toilet tank, the pessue gadient in the pipes fo cold wate dops, deceasing the flow of cold wate and exposing you to the hot wate alone ouch!! Howeve, even in the moe geneal case of the Navie-Stokes equation that has an inetial tem, ρ u z + uz t z, one can see that fo steady flow u z the geomety of the poblem and the t incompessibility of the fluid specify that the inetial tem is exactly zeo. So Poiseuille Flow is not limited to the Stokes egime, but also occus at highe Re and we ll see that this is impotant. This 1-D vesion of the momentum equation in cylindical coodinates is then We will ty a solution of the fom dp + η u z o dp + η 1 u z = 1 dp 4η + c 1 ln + c u z 1 subject to the bounday conditions of no-slip side walls and finite foce ove the fluid length u z at u z at = a 3 Solving fo the constants we now have u z = 1 dp 4η a 4 This means the velocity pofile of the flow has a paabolic shape with a maximum in the cente and is zeo at the pipe walls. Also note that the flow is independent of the fluid density. 1

2 As mentioned, the velocity is maximum at the cente, which we can calculate u max = dp a 4η at 5 Pessue gadients ae nomally defined to be negative, such that wate flows fom high pessue to low pessue, so when P 1 > P 0, u max is a positive quantity. It is also useful to calculate the total flow ate though the pipe, so we integate the velocity ove the a coss-section of the pipe Q = a 0 π ud = a 0 π 1 dp 4η a d = dp πa 4 8η = πa4 P 0 P 1 8ηl 6 The volumetic flow ate units of volume/time o m 3 /s shows that fo a given pessue gadient and viscosity, the flow though the pipe is popotional to the adius of the pipe to the fouth powe. This is what Poiseuille demonstated expeimentally. The mean velocity is simply the total flow nomalized by the coss-sectional aea of the pipe ū = dp πa 4 8πa η = dp a 8η = 1 u max 7 The mean velocity is the esult of the net foce exeted on the fluid by the pessue gadient acting to ovecome the viscous dag fom the pipe walls. The foce pe unit length fom pessue is F P = πa P 0 P 1 l = πa dp This shows that the mean flow, ū, is elated to the pessue foce by ū = F P /8πη and so it is linealy invesely popotional to η. Similaly, fo a Newtonian fluid, viscous dag is popotional to the shea tangential stess, σ z, which we can evaluate nea the wall of the pipe, = a σ z =a = η u z =a = η a dp η = a dp Simila to the non-dimensional dag coefficient of the Stokes sphee, c D, we can detemine a fiction facto, f, which descibes the effect of dag. We use the shea stess evaluated at the wall as a chaacteistic stess and nomalize that value by a chaacteistic pessue 1ρ fū in which we use the mean velocity f = σ z =a 1 ρ fū = 4a dp 10 ρ f ū If we substitute fo just one of the ū, then f looks like f = 4a 1 dp ρ f ū ū = 4a ρ f ū 8η a dp dp = 3η ρ f ūa If we choose a chaacteistic length scale as the diamete of the pipe, D = a, then we have f = 64η ρ f ūd = 64 Re 1 This elationship holds until the tansition into the tubulent flow egime at Re

3 Channel Flow Anothe unidiectional flow is the flow between two igid plates diven by a pessue gadient. This is actually just Poiseuille flow in Catesian geomety with ẑ the same diection as in cylindical pola so the pessue gadient and esultant flow ae both only in the x diection u y = u z and the velocity pofile vaies with z. The geomety has the x-axis along the mid-plane of the channel, and since the channel has height h, the channel walls ae at ±h/. The govening equations ae u x x dp + dx η u x o dp + η u x dx z Since dp/dx is constant, this is a second ode O.D.E. and the integation is staightfowad 13 u x = 1 dp η dx z + c z + c 1 14 The bounday conditions ae fom the mio symmety along the mid-plane u x z = u x z and no-slip at the walls u x z = ±h We can now solve fo the constants of integation and get the velocity pofile. All the same insights fom Poiseuille flow in a pipe ae applicable hee. u x = 1 dp [ z h/ ] 15 η dx The velocity pofile is again paabolic in shape and constant eveywhee. Couette Flow Couette flow is simila to channel flow and has the same geomety but with an impotant modification. Instead of the pessue gadient diving the flow, it is diven by the motion of one of the boundaies and that motion is paallel to the diection of the channel dp is in fact absent fom dx this poblem. The assumption is that some extenal foce is applied to move the wall and that applied foce simply scales with the viscosity of the fluid. Depending on the efeence fame you choose to do the poblem in, the top o bottom plate can be moving at some velocity U 0 o they can both move in opposite diections at U 0 /. The most convenient choice fo the coodinate system is to have a stationay plate at z and a moving plate at z = h so again the channel has height h. The govening equations fo a shea diven flow ae even simple than fo channel flow since now dp dx u x x 16 0 = η u x o 0 = η u x z Twice integating this second ode O.D.E. gives the solution u x = c 1 z + c. The bounday conditions ae again no-slip velocity bounday conditions at the stationay and moving walls, so u x z and u x z = h = U 0. The solution fo the u x is again a constant z u x = U 0 17 h The velocity pofile in a shea diven flow is again identical fo all values of x, vaies linealy with distance fom the moving wall, and is independent of both density and viscosity. Also note that the shea stess is also constant eveywhee σ xz = η u x z = η U 0 h 18 3

4 Classification of PDEs and types of Bounday Conditions Any PDE can be classified using the method of chaacteistics which detemines if the PDE is eithe hypebolic, elliptic, o paabolic. Both Laplace s equation and Poisson s equation ae classified as elliptical, and is a common class of equation one encountes in fluid dynamics. Othe examples include of the wave equation hypebolic and the diffusion equation paabolic. It is impotant to undestand which class of equation you ae attempting to solve, in paticula if you ae using numeical methods, because the stability o success of the numeical method applied to one class of equation may be a completely unstable o be an unsuccessful appoach if applied to a diffeent class of PDE. The pimay vaiable is the vaiable in the govening equation eithe PDE o ODE and evey pimay vaiable always has an associated seconday vaiable. The seconday vaiable is usually the deivative of the pimay vaiable and always has a physical meaning that is often a quantity of inteest. In fluid dynamics the pimay vaiable is velocity and the seconday vaiable is stess. Anothe example is heat tansfe in which the pimay vaiable is tempeatue and the seconday vaiable is heat flux. In ode to obtain a solution to any PDE, bounday conditions must be specified. Thee ae two types of bounday conditions that can be applied: those that specify the pimay dependent vaiable on bounday and those that specify a seconday vaiable on the bounday, and usually the deivative is taken nomal to the bounday. The fist type of bounday condition is called an essential bounday condition and when solving an elliptic class of equation it is known as a Diichlet bounday condition. The second type of bounday condition is called a natual bounday condition and when solving an elliptic class of equation it is known as a Neumann bounday condition. It is quite ok, and even somewhat common, to have mixed types of bounday conditions along diffeent pats of the bounday. Fo example, one potion of the bounday will specify a Diichlet bounday condition and anothe potion will specify a Neumann bounday condition. Howeve, it is impossible to specify both types of conditions at the same point of any potion of the bounday. Thus, if the tempeatue is specified, the heat flux will be detemined o vice-vesa but it can neve happen that both ae specified at the same place. Similaly, if the stess is specified at a given point, the velocity will be solved fo on the bounday at the same point. This is actually quite a poweful, and useful, thing to know, especially in situations like Couette flow and channel flow, which have the same geomety. It is actually possible to combine the simple solutions fom both poblems because 1 they ae both linea ODEs we can use the pinciple of supeposition and the solutions wee aived upon by applying the same type of bounday condition. Both poblems specified the velocity on the walls and theefoe both applied Diichlet bounday conditions. We can then wite the solution of Couette flow that now includes a pessue gadient by simply tansfoming the channel flow solution to a coodinate system with the bottom wall at z so z u x = U 0 h + 1 dp [ z hz ] 19 η dx A simple model of asthenospheic counteflow is motivated by a shea flow diven by plate motions on the suface. The shea flow sets up a pessue gadient in the the opposite diection which dives an associated channel flow undeneath the shea flow a etun flow. This is the same as the above poblem, except the diection of the pessue gadient is evesed z u x = U 0 h 1 dp [ z hz ] 0 η dx 4

5 Steam Function The steam function, ψ, is both an illustative and useful appoach to apply to fluid dynamics as it can povide elatively quick solutions to -D incompessible flow poblems. The majo dawback of the steam function is that it is basically limited entiely to -D incompessible flow poblems. The steam function is like a potential field in that only the diffeence in ψ between two points has any physical meaning the absolute value of ψ is abitay. Lines of constant ψ ae called steam lines and give an excellent visual epesentation of the flow, howeve, only in a -D geomety. In -D, the incompessibility constaint is fom the continuity equation The definition of the steam function is then u o u x x + u y y 1 u x = ψ y u y = ψ x The steam function satisfies the continuity equation ψ x y + ψ y x 3 The steam function can also be substituted into the Stokes equation 0 = dp η 3 ψ + 3 ψ dx x y 3 y 0 = dp dy + η 4 3 ψ + 3 ψ 3 x y x Now eliminate the pessue tem using the same technique that was applied ealie when solving fo the flow aound a Stokes sphee, i.e. take patial deivatives w..t. the othe dimension [ ] 0 = dp η 3 ψ + 3 ψ y dx x y 3 y [ ] 5 0 = dp + η 3 ψ + 3 ψ x dy 3 x y x and then subtacting the esulting equations we get 0 = 4 ψ 4 x + 4 ψ x y + 4 ψ 4 y Reaanging the deivatives we now have 0 = x + y x + y 6 ψ 7 This equation can be ecognized as the Laplacian opeato being applied twice to ψ 0 = ψ 8 And this is known as the Bihamonic opeato = 4 which we can use to wite 0 = 4 ψ 9 Thee ae well-known solutions to this equation and it is also valid fo non-catesian geometies. 5

6 Cone Flow The situation of a subduction zone is in some ways analogous to one vaiation of the classic cone flow poblem in fluid dynamics. In this vesion, two igid plates infinite in extent convege at a point whee the advancing plate plate A dips at an angle below the back-ac plate plate B. We will use the point of convegence as the oigin of a -D cylindical coodinate system with plates on the suface the line at θ. The angle that plate A makes between itself on the suface and the dipping potion is defined as θ a and the dip angle between plates A and B is defined as θ b and assumed to be acute. Plate B is assumed to emain stationay while plate A is moving on the suface at velocity u = U 0 towads the oigin and along the dip angle at u = U 0 away fom the oigin. Fo both plates, the velocities in the θ diection ae assumed to be zeo u θ. Notice that thee ae no body foces in this poblem, and that the Stokes flow is diven entiely by the velocity bounday conditions which themselves ae diving by some applied foce but since it is not a body foce it is ielevant. The govening equations fo Stokes flow ae simply τ and u. Expanding the momentum equation out into the components of total stess τ τ θ τ θ θ τ θθ θ We can use the constitutive elationship between total stess and stain ate, τ = P I + ηd 30 τ = P + σ = P + η ε τ θθ = P + σ θθ = P + η ε θθ 31 τ θ = τ θ = σ θ = η ε θ And ewite the total stess with the stain ate having tems of the velocity gadients τ = P + η u τ θθ = P + η 1 u θ θ τ θ = η 1 u θ + u θ + u u θ Notice that if we add the nomal components of stess togethe we get u τ + τ θθ = P + η + 1 u θ θ + u The nd tem on the RHS vanishes since u, and because the fluid is isotopic the expessions fo nomal stesses become τ = P τ θθ = P Using these allows us to expess the momentum equation entiely in tems of P and τ θ P + 1 τ θ θ 1 P + τ θ θ 35 6

7 and this can be ewitten as P + η 1 1 P θ + η 1 u θ θ 1 u θ + u θ + u θ u θ u θ 36 Seems all that manipulation didn t help as the momentum equation still looks a little ugly. Luckily, it was shown that solving 4 ψ will also give the solution fo velocity, and it tuns out the steam function is a moe convenient way to appoach the poblem. In -D cylindical, the Laplacian is ψ = 1 ψ + 1 ψ 37 θ Consideing the geomety of the poblem has plates of infinite extent with constant elative velocity, the solution fo velocity eveywhee is expected to be independent of. This means the equation is sepaable and we will use a solution of the fom Simple substitution fo ψ gives ψ = RT θ and 38 u = 1 ψ θ u θ = ψ u = 1 θ R T 40 θ which means R = and then ψ = T θ which upon substituting back into the Bihamonic equation gives 4 T θ + T 4 θ + T 41 The 4 th ode PDE has now been educed to a 4 th ode ODE which has a geneal solution of the fom T θ = A sin θ + B cos θ + Cθ sin θ + Dθ cos θ 4 and thee ae also 4 bounday conditions but these ae given as velocities so we need u and u θ 39 u = T θ θ = A cos θ B sin θ + Csin θ + θ sin θ + Dcos θ θ sin θ u θ = T θ 43 At this point its a good idea to beak the poblem into two potions and solve fo the steam function in each domain. The obvious choice fo the two domains is the back-ac egion fomed by the acute dip angle between the subducting plate and oveiding plate and the foe-ac egion undeneath the subducting plate. The flows ae identical along the bounday of the subducting plate, and this line is known as the sepaatix. The bounday conditions ae then u θ = U 0 in the foe-ac egion u θ in the back-ac egion u θ θ in both egions u θ θ = θ b along the sepaatix u θ θ = θ b = U 0 along the sepaatix 44 7

8 Each egion has 4 bounday conditions to solve fo the 4 unknowns constants, and afte a lot of algeba one aives at the solution ψ a = U 0[θ a θ sin θ θ sinθ a θ] θ a+sin θ a ψ b = U 0[θ b θ sin θ b sin θ θ b θ sinθ b θ] θ b sin θ b o moe simply, ψ a = U 0 f a θ in the foe-ac egion o moe simply, ψ b = U 0 f b θ in the back-ac egion The velocities in each egion ae eadily obtained though diffeentiation of ψ: u = U 0 f aθ and u θ = U 0 f a θ in the foe ac and u = U 0 f b θ and u θ = U 0 f b θ in the back ac. In ode to obtain the pessue, we need to go back to the momentum equation and use the fact that τ = τ θθ = P which itself is elated to the shea tangential stess τ θ = η U 0 [ f a θ f a θ] τ θ = η U 0 [f a θ + f a θ] This helps obtain the pessue solution P a, θ = U 0η P b, θ = U 0η [sin θ sinθ θ a] θ a+sin θ a [θ b sinθ b θ sin θ b sin θ] θ b sin θ b in the foe-ac egion in the back-ac egion in the foe-ac egion in the back-ac egion Inspection of these solutions eveals that P a is always a positive quantity and P b is always a negative quantity. A positive pessue below the subducting plate implies compession o upwad foce on the suface. A negative pessue in the mantle wedge indicates that thee is a suction between the subducting plate and oveiding plate. This cone flow suction acts as a hydodynamic lift that is popotional to the pessue diffeence above and below the slab. The lift is found by integating P, θ along the dip angle, θ a, ove a length l. The toque exeted by lift is balanced by gavity though the weight of the slab with thickness h and density ρ. T flow = [ ] l [P sin θ 0 a, θ a P b, θ b ] d = U 0 ηl b π θ b +sin θ b + sin θ b θb sin θ b 48 T gavity = 1 ρghl cos θ b Both toques can be nomalized by a chaacteistic toque, U 0 ηl, which then allows one to find the citical dip angle, θ c that detemines when the toque deived fom gavity is balanced by the lift geneated by ciculation in the mantle wedge. Fo any angle smalle than θ c, the toque exeted on the slab by mantle flow will exceed the weight of the slab, and assuming the velocities emain constant, a positive feedback will occu such that θ deceases to zeo. This citical angle was detemined by Stevenson and Tune, 1977, to be 63 fo which they found the net toque was about times the chaacteistic toque. Assuming a 100 km thick slab that is 600 km in length subducts at 6 cm/y and has ρ =80 kg/m 3 gives ρghl 4ηU 0 which can be used to estimate the uppe mantle viscosity, η = π 10 1 Pa s 49 Clealy, θ c = 63 is too lage as many slabs ae obseved to have dip angles shallowe than this estimate, so obviously thee must be many othe impotant factos. One of the moe impotant factos is the non-newtonian heology of the mantle wedge as studied by Tovish et al who found this educed θ c = 54 fo a powe law fluid with n=3. Thee ae also easons fo θ c to be lage, as slabs with finite lateal extent allow fo a 3-D component of the mantle flow i.e the tooidal flow aound slab edges which educes the pessue diffeential Dvokin et al

### Introduction to Fluid Mechanics

Chapte 1 1 1.6. Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = 3.174 ft/s. (a) What is its mass in kg? (b) What will the weight of this body

More information

### Gauss Law. Physics 231 Lecture 2-1

Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

### Physics 235 Chapter 5. Chapter 5 Gravitation

Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

More information

### 2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

More information

### 12. Rolling, Torque, and Angular Momentum

12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

More information

### FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

More information

### Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

More information

### Gravitation. AP Physics C

Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

More information

### Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

More information

### Fluids Lecture 15 Notes

Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2-D, this velocit

More information

### VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

More information

### Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

More information

### The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C

Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit

More information

### Voltage ( = Electric Potential )

V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

More information

### Voltage ( = Electric Potential )

V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

More information

### Chapter 4: Fluid Kinematics

Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian

More information

### Carter-Penrose diagrams and black holes

Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

More information

### Mechanics 1: Motion in a Central Force Field

Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

More information

### PY1052 Problem Set 8 Autumn 2004 Solutions

PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

More information

### Chapter 2. Electrostatics

Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.

More information

### Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,

More information

### Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

### Problem Set # 9 Solutions

Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

More information

### Deflection of Electrons by Electric and Magnetic Fields

Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

More information

### Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

More information

### Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3

Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each

More information

### Structure and evolution of circumstellar disks during the early phase of accretion from a parent cloud

Cente fo Tubulence Reseach Annual Reseach Biefs 2001 209 Stuctue and evolution of cicumstella disks duing the ealy phase of accetion fom a paent cloud By Olusola C. Idowu 1. Motivation and Backgound The

More information

### Experiment 6: Centripetal Force

Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

More information

### Chapter 4: Fluid Kinematics

4-1 Lagangian g and Euleian Desciptions 4-2 Fundamentals of Flow Visualization 4-3 Kinematic Desciption 4-4 Reynolds Tanspot Theoem (RTT) 4-1 Lagangian and Euleian Desciptions (1) Lagangian desciption

More information

### Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

More information

### Coordinate Systems L. M. Kalnins, March 2009

Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

### CHAPTER 10 Aggregate Demand I

CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

More information

### Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

More information

### Solutions for Physics 1301 Course Review (Problems 10 through 18)

Solutions fo Physics 1301 Couse Review (Poblems 10 though 18) 10) a) When the bicycle wheel comes into contact with the step, thee ae fou foces acting on it at that moment: its own weight, Mg ; the nomal

More information

### Chapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom

Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in

More information

### Forces & Magnetic Dipoles. r r τ = μ B r

Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

More information

### 2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

More information

### Physics HSC Course Stage 6. Space. Part 1: Earth s gravitational field

Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe

More information

### Exam 3: Equation Summary

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

More information

### Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

More information

### 4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

More information

### Experiment MF Magnetic Force

Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuent-caying conducto is basic to evey electic moto -- tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating

More information

### Lab #7: Energy Conservation

Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

More information

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

More information

### An Introduction to Omega

An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

More information

### PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0

More information

### Financing Terms in the EOQ Model

Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad

More information

### Continuous Compounding and Annualization

Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

More information

### Chapter 30: Magnetic Fields Due to Currents

d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.1-1 T) i to ue a lage cuent flowing though a wie.

More information

### Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem

Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the

More information

### Determining solar characteristics using planetary data

Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation

More information

### TECHNICAL DATA. JIS (Japanese Industrial Standard) Screw Thread. Specifications

JIS (Japanese Industial Standad) Scew Thead Specifications TECNICAL DATA Note: Although these specifications ae based on JIS they also apply to and DIN s. Some comments added by Mayland Metics Coutesy

More information

### Ilona V. Tregub, ScD., Professor

Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

More information

### Lab M4: The Torsional Pendulum and Moment of Inertia

M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disk-like mass suspended fom a thin od o wie. When the mass is twisted about the

More information

### (a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of

Homewok VI Ch. 7 - Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the

More information

### Problems of the 2 nd and 9 th International Physics Olympiads (Budapest, Hungary, 1968 and 1976)

Poblems of the nd and 9 th Intenational Physics Olympiads (Budapest Hungay 968 and 976) Péte Vankó Institute of Physics Budapest Univesity of Technology and Economics Budapest Hungay Abstact Afte a shot

More information

### Chapter 2 Modelling of Fluid Flow and Heat Transfer in Rotating-Disk Systems

Chapte 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems 2.1 Diffeential and Integal Equations 2.1.1 Diffeential Navie Stokes and Enegy Equations We will conside hee stationay axisymmetic

More information

### Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

More information

### Gravity. A. Law of Gravity. Gravity. Physics: Mechanics. A. The Law of Gravity. Dr. Bill Pezzaglia. B. Gravitational Field. C.

Physics: Mechanics 1 Gavity D. Bill Pezzaglia A. The Law of Gavity Gavity B. Gavitational Field C. Tides Updated: 01Jul09 A. Law of Gavity 3 1a. Invese Squae Law 4 1. Invese Squae Law. Newton s 4 th law

More information

### The LCOE is defined as the energy price (\$ per unit of energy output) for which the Net Present Value of the investment is zero.

Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the

More information

### It is required to solve the heat-condition equation for the excess-temperature function:

Jounal of Engineeing Physics and Themophysics. Vol. 73. No. 5. 2 METHOD OF PAIED INTEGAL EQUATIONS WITH L-PAAMETE IN POBLEMS OF NONSTATIONAY HEAT CONDUCTION WITH MIXED BOUNDAY CONDITIONS FO AN INFINITE

More information

### AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,

More information

### Charges, Coulomb s Law, and Electric Fields

Q&E -1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded

More information

### The Binomial Distribution

The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

More information

### Uniform Rectilinear Motion

Engineeing Mechanics : Dynamics Unifom Rectilinea Motion Fo paticle in unifom ectilinea motion, the acceleation is zeo and the elocity is constant. d d t constant t t 11-1 Engineeing Mechanics : Dynamics

More information

### CHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL

CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The

More information

### The Gravity Field of the Earth - Part 1 (Copyright 2002, David T. Sandwell)

1 The Gavity Field of the Eath - Pat 1 (Copyight 00, David T. Sandwell) This chapte coves physical geodesy - the shape of the Eath and its gavity field. This is just electostatic theoy applied to the Eath.

More information

### Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning

Gavitational Mechanics of the Mas-Phobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This

More information

### 1D STEADY STATE HEAT

D SEADY SAE HEA CONDUCION () Pabal alukda Aociate Pofeo Depatment of Mecanical Engineeing II Deli E-mail: pabal@mec.iitd.ac.in Palukda/Mec-IID emal Contact eitance empeatue ditibution and eat flow line

More information

### The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

More information

### DYNAMICS AND STRUCTURAL LOADING IN WIND TURBINES

DYNAMIS AND STRUTURAL LOADING IN WIND TURBINES M. Ragheb 12/30/2008 INTRODUTION The loading egimes to which wind tubines ae subject to ae extemely complex equiing special attention in thei design, opeation

More information

### F G r. Don't confuse G with g: "Big G" and "little g" are totally different things.

G-1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just

More information

### STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

More information

### Analytical Proof of Newton's Force Laws

Analytical Poof of Newton s Foce Laws Page 1 1 Intouction Analytical Poof of Newton's Foce Laws Many stuents intuitively assume that Newton's inetial an gavitational foce laws, F = ma an Mm F = G, ae tue

More information

### TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION

MISN-0-34 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................

More information

### Magnetic Bearing with Radial Magnetized Permanent Magnets

Wold Applied Sciences Jounal 23 (4): 495-499, 2013 ISSN 1818-4952 IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.2013.23.04.23080 Magnetic eaing with Radial Magnetized Pemanent Magnets Vyacheslav Evgenevich

More information

### Quantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w

1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a

More information

### Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

More information

### www.sakshieducation.com

Viscosity. The popety of viscosity in gas is due to ) Cohesive foces between the moecues ) Coisions between the moecues ) Not having a definite voume ) Not having a definite size. When tempeatue is inceased

More information

### The Role of Gravity in Orbital Motion

! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

More information

### Classical Mechanics (CM):

Classical Mechanics (CM): We ought to have some backgound to aeciate that QM eally does just use CM and makes one slight modification that then changes the natue of the oblem we need to solve but much

More information

### Converting knowledge Into Practice

Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading

More information

### 2. Orbital dynamics and tides

2. Obital dynamics and tides 2.1 The two-body poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body

More information

### Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

More information

### AP Physics Electromagnetic Wrap Up

AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle

More information

### Displacement, Velocity And Acceleration

Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,

More information

### Semipartial (Part) and Partial Correlation

Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

More information

### Pessu Behavior Analysis for Autologous Fluidations

EXPERIENCE OF USING A CFD CODE FOR ESTIMATING THE NOISE GENERATED BY GUSTS ALONG THE SUN- ROOF OF A CAR Liang S. Lai* 1, Geogi S. Djambazov 1, Choi -H. Lai 1, Koulis A. Peicleous 1, and Fédéic Magoulès

More information

### 3.02 Potential Theory and Static Gravity Field of the Earth

3.02 Potential Theoy and Static Gavity Field of the Eath C. Jekeli, The Ohio State Univesity, Columbus, OH, USA ª 2007 Elsevie B.V. All ights eseved. 3.02. Intoduction 2 3.02.. Histoical Notes 2 3.02..2

More information

### Construction of semi-dynamic model of subduction zone with given plate kinematics in 3D sphere

Eath Planets Space, 62, 665 673, 2010 Constuction of semi-dynamic model of subduction zone with given plate kinematics in 3D sphee M. Moishige 1, S. Honda 1, and P. J. Tackley 2 1 Eathquake Reseach Institute,

More information

### 10. Collisions. Before During After

10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: -F(t)

More information

### 7 Circular Motion. 7-1 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary

7 Cicula Motion 7-1 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o

More information

### Strength Analysis and Optimization Design about the key parts of the Robot

Intenational Jounal of Reseach in Engineeing and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Pint): 2320-9356 www.ijes.og Volume 3 Issue 3 ǁ Mach 2015 ǁ PP.25-29 Stength Analysis and Optimization Design

More information

### Solution Derivations for Capa #8

Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass

More information

### SELF-INDUCTANCE AND INDUCTORS

MISN-0-144 SELF-INDUCTANCE AND INDUCTORS SELF-INDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. Self-Inductance L.........................................

More information

### ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS

ON THE R POLICY IN PRODUCTION-INVENTORY SYSTEMS Saifallah Benjaafa and Joon-Seok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poduction-inventoy

More information

### The Supply of Loanable Funds: A Comment on the Misconception and Its Implications

JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently Fields-Hat

More information

### A r. (Can you see that this just gives the formula we had above?)

24-1 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down - you can pedict (o contol) motion

More information

### Saving and Investing for Early Retirement: A Theoretical Analysis

Saving and Investing fo Ealy Retiement: A Theoetical Analysis Emmanuel Fahi MIT Stavos Panageas Whaton Fist Vesion: Mach, 23 This Vesion: Januay, 25 E. Fahi: MIT Depatment of Economics, 5 Memoial Dive,

More information

### How To Find The Optimal Stategy For Buying Life Insuance

Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,

More information